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E. A. Emerson and C. S. Jutla, Tree Automata, Mu-Calculus, And

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E. A. Emerson and C. S. Jutla, Tree Automata, Mu-Calculus, And Tree Automata MuCalculus and Determinacy Extended Abstract EA Emerson and CS Jutla The University of Texas at Austin IBM TJ Watson Research Center Abstract that tree automata are closed under disjunction pro jection and complementation While the rst two We show that the prop ositional MuCalculus is eq are rather easy the pro of of Rabins Complemen uivalent in expressivepower to nite automata on in tation Lemma is extraordinarily complex and di nite trees Since complementation is trivial in the Mu cult Because of the imp ortance of the Complemen Calculus our equivalence provides a radically sim tation Lemma a numb er of authors have endeavored plied alternative pro of of Rabins complementation and continue to endeavor to simplify the argument lemma for tree automata which is the heart of one HR GH MS Mu Perhaps the b est of the deep est decidability results We also showhow known of these is the imp ortant work of Gurevich MuCalculus can b e used to establish determinacy of and Harrington GH which attacks the problem innite games used in earlier pro ofs of complementa from the standp oint of determinacy of innite games tion lemma and certain games used in the theory of While the presentation is brief the argument is still online algorithms extremely dicult and is probably b est appreciated Intro duction y the page supplementofMonk when accompanied b Mon We prop ose the prop ositional Mucalculus as a uniform framework for understanding and simplify In this pap er we present a new enormously sim ing the imp ortant and technically challenging areas plied pro of of the Complementation Lemma We of automata on innite trees and determinacy of in would argue that the new pro of is straightforward and nite games We show that the MuCalculus is pre natural To obtain our pro of weshow that the prop o cisely equivalent to tree automata in expressivepower sitional MuCalculus Ko is expressively equiva p ermitting a radically simplied pro of of the Comple lent to nondeterministic tree automata Since the mentation Lemma for tree automata We also show MuCalculus is trivially closed under complementa how to systematically prove the determinacy of cer tion the equivalence immediately implies that tree tain innite games including games in the theory of automata are also closed under complementation online algorithms We remark that the equivalence of the Mucalculus Rabin intro duced tree automata nite state and tree automata is a new result of some indep en automata on innite trees to prove the decidabil dentinterest since it shows that there indeed exists a ity of the monadic second order theory of n succes natural mo dal logic of programs precisely equivalent sors SnS This is one of the most fundamental de to tree automata in expressive power cf Th cidability results to which many other decidability Actually a brief argument establishing this equiva results in logic mathematics and computer science lence can be given by app ealing to known lengthy can be reduced The pro of involves reducing satis arguments translating through SnS However a care ability of SnS formulae to the nonemptiness prob ful examination of this argumentreveals that rep eated lem of tree automata The reduction entails showing app eals to the Complementation Lemma are involved Thus this argument is not at all suitable for use in a us to avoid the app eal to GH in our translation pro of of the Complementation Lemma The relation That proves the equivalence of nondeterministic Tree ship of the MuCalculus and tree automata has also Automata and the MuCalculus and hence the Com been investigated previously by Niwinski In Ni plementation Lemma it is shown that tree automata can b e translated into Almost all published pro ofs of the Complementa the MuCalculus while in Ni it is shown that a re tion Lemma RaBu GH can be seen stricted MuCalculus in which conjunctions are not up on reection to havetaken the following approach allowed can b e translated into tree automata As we b est broughtoutby MS shall see in our translation of the full MuCalculus Alternating Tree Automata are easy to comple into tree automata the main diculty lies in dealing ment given the fact that certain innite games with the conjuncts are determinedie at least one of the play ers has a The MuCalculus L is a mo dal logic we deal winning strategy Nondeterministic Tree Automata here with the prop ositional Temp oral version with clearly are alternating Tree Automata Alternating xp oint op erators It provides a least xpoint operator Tree Automata can be reduced to nondeterministic and a greatest xpoint operator which makes Automata as shown in this pap er using determiniza it p ossible to give extremal xp ointcharacterizations tion of Automata and aforgetful strategy theo of the branching time mo dalities Intuitiv rem ely the Mu Calculus makes it p ossible to characterize the mo dal Theorems and in all the results mentioned ities in terms of recursively dened treelike patterns ab ove have had dicult pro ofs Although GH First we show that the parse tree of a formula of have somewhat simplied the pro ofs over Ra the L can b e viewed as an alternating Tree Automaton pro ofs are far from b eing transparent Moreover their MS giving a translation from L to alternating pro ofs of Theorem and are intertwined The Tree Automata The problem of translating the Mu new pro of of the Complementation Lemma via Mu Calculus into nondeterministic Tree Automata then Calculus suggests the following for simplifying the reduces to translating alternating Tree Automata to ab ove approach ie and as well To prov e nondeterministic Tree Automata extremal xp oints Mucalculus can b e used to The alternating Tree Automata obtained from the characterize games in which a particular Player has a MuCalculus however have a nice prop erty in the winning strategy Indeed in section we give simple sense that if there is an accepting run of an automa and straightforward Mucalculus characterizations of ton A on an input tree t then there is a historyfree games in which Players I and I I have winning strate accepting run of A on t A run of A on t is historyfree gies which turn out to b e trivial complements of each i the nondeterministic choices made by A along any other Moreover a totally indep endent ranking argu t state ie the path of t dep end only on the curren mentprovides a simple pro of of choices are indep endent of the history of the dierent Finallywealsoshow that the ab ovetechnique of forallruns extremal xp oint Mucalculus characterization gives Weshowhow these historyfree alternating Tree a general metho d for proving various determinacy re Automata can be reduced to nondeterministic Tree sults Wolfe cf Mo proved the deter Automata using constructions for determinizingSa minacy of innite games in the second level of the and complementing EJ automata on innite strings Borel hierarchyF Weshowthatitispossibleto For general alternating tree automata we can still do view such determinacy results as implicitly construct the translation into nondeterministic tree automata ing extremal xp oints MuCalculus characterization but nowwemust use Gurevich and Harringtons for also gives a clear and simple pro of of determinacy of getful strategy theorem GH the historyfreedom sp ecialized F games used in derandomizing exis of the automata derived from the MuCalculus allows tentially Online algorithms RS This makes the sequence of OR no des s s s visited along b in the result in RS only slightly more complicated b eing k play which we dene by a function path Strat more general than a determinacy result in B I f g OR Detailed pro ofs of Theorems in section and Similarly given app ear in the second authors dissertation J b f g and a legal strategy there is also a unique innite sequence of OR I Alternating Tree Automata to Nondeter no des visited along b if Player I follows Thus I ministic Tree Automata we dene a function ipath Strat f g OR I We deal here only with nite state Automata with ipath bs s OR where b b b I such that s b and k s Denition A nondeterministic binary TreeAu I k b b b b Note that fpath b jb pre tomaton is a tuple N OR AN D I k k I x of bg is exactly the set of nonempty nite prexes Ls where of ipath b is the input alphab et I OR is the nite set of states The acceptance condition is given by a sub AN D is a set of no des intuitively corresp onding to set of OR We say that x OR satises i entries in the transition function of the Automaton x is in the subset representing Usually is de L AN D is a lab elling of the AN D no des with ned by a Temp oral formula interpretted over in the input symb ols nite sequences of OR no des For example the Streett is a relation on OR AN D dening the dierent acceptance condition has m pairs of subsets of OR nondeterministic choices from OR to AN D no des es fGR E E N RED GR E E N RED g Then i m m V sentially the transition function the subset of OR dening is given by GF im AN D f g OR is a function sp ecifying left GR E E N GF R E D In Temporal Logic G stands i i and right successor states for everywhere and F stands for eventually The s OR is the start state Pairs acceptance condition is the complementofthe is the acceptance condition to b e dened b elow Streett condition Wenowgive a game theoretic denition of acceptance Denition A legal strategy forIisawinning I of a tree t by an Automaton N strategy for I i b f g ipath b satises Denition Given an input tree t f g I Wesaythat N accepts t i I has
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